2020/04/15 by Federico Castillo, Jeremy E. Martin, Castillo, Federico +3 · 1 citation
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2004.07308
openalex publication_date 2020/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study ordered matroids and generalized permutohedra from a Hopf theoretic point of view. Our main object is a Hopf monoid in the vector species of extended generalized permutahedra equipped with an order of the coordinates; this monoid extends the Hopf monoid of generalized permutahedra studied by Aguiar and Ardila. Our formula for the antipode is cancellation-free and multiplicity-free, and is supported only on terms that are compatible with the local geometry of the polyhedron. Our result is part of a larger program to understand orderings on ground sets of simplicial complexes (for instance, on shifted and matroid independence complexes). In this vein, we show that shifted simplicial complexes and broken circuit complexes generate Hopf monoids that are expected to exhibit similar behavior.